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55/1970

Staff

Full time

THE AUSTRALIAN NATIONAL UNIVERSITY

SCHOOL OF GENERAL STUDIES

FACULTY OF ARTS

DEPARTMENT OF PURE MATHEMATICS

ANNUAL RE PORT 1969

Professor: Hanna Neumann, F.A.A., D.Phil., D.Sc.(Oxon). Reader: M.F. Newman, M.Sc.(Syd), Ph.D.(Manc). Visiting Senior Lecturer: J. Wiegold, M.Sc.(Manc), Ph.D.(Manc). Senior Lecturer: I.M.S. Dey, B.Sc., M.Sc.Tech., Ph.D.(Manc) (since 1.7.69). Lecturers: c. Christensen, M.Sc.Tech., Ph.D.(Manc).

P.J. Cossey, B.Sc.(Qld), Ph.D.(ANU). T.K. Donaldson, B.S.(Kentucky), M.s., Ph.D.(Chic) (since 5.6.69). A. Howe, B.Sc.(N.E.), Ph.D.(ANU) (since 25.8.69). W.L. Steiger, S.M.(M.I.T), Ph.D.(ANU) (resigned 1.9.69). M.A. Ward, B.Sc.(Melb), B.A.(ANU), Ph.D.(ANU).

Senior Tutor: N.D. Porter, B.Sc.(Melb), M.Sc.(ANU). Tutor: L.M. Butler, B.Sc.(Qld), M.Sc.(ANU). Temporary Tutor: P.F. Pentony, B.Sc(ANU) (from 8.9.69 to 28.11.69). Clerk: Mrs F.A. Wickland

Part time Tutors: A. Isaacs

K. Palmer (1st and 2nd terms only)

" II " P. F. Pentony 11 11 11

J.M. Brady (1st, 2nd and 3rd terms) E .R. Coleman " " 11 11

J.R.J. Groves 11

w. Haebich " J.A. McLucas "

II

II

" J.F. Price " 11

D.G. Tacon " 11

C.P. Vlagsma " 11

Dr Christine Wiegold Typist: Mrs A. Zalucki (since

" II

II II

" II

II II

II II

" " " II

6.1.69).

There have been some staff movements. Dr Steiger has returned to the U.S.A.; Drs Donaldson and Howe have come from North America (though the latter is returning to Australia). Both Donaldson and Howe are young analysts. This should relieve somewhat the pressure on other members of staff to teach outside their normal fields of activity. However the need for a topologist and for senior staff continues.

Miss Butler took leave for third term and was ably replaced by Mr Pentony who took leave from his Ph.D. course.

Professor Neumann was elected to the Australian Academy of Science. Dr Dey has been promoted to a Senior Lectureship. Dr Steiger was awarded a Ph.D. for a thesis submitted in the Department of Statistics (RSSS). Miss Butler was awarded an M.Sc.

Dr Steiger was on study leave in North America during first term; while there he gave seminars at the University of Toronto and Carnegie-Mellon University. Professor Neumann commenced a year's study leave in August. She is spending the first part of this at Vanderbilt University (Tennessee) on a National Science Foundation (NSF) Senior Foreign Science Fellow grant after attending the ler Congres International de 1 1Enseignement Math~matique at Lyon and a conference on decision problems in Group Theory in Los Angeles.

Teaching

Undergraduate:

2.

55/1970

There is a strong feeling among the staff that the present student­staff ratio (effectively 14 to 1) will hamper further developments in undergraduate teaching for which there is a clear need. In fact it may well lead to some deterioration from the position achieved to date because there is a need to balance inunediate demands against long-term ones which require that the staff devote a reasonable amount of time to their scholarly work. The problem is aggravated by the imbalance between junior and senior staff. In this respect the absence of Professor Neumann on study leave is showing the rest of us that we should be grateful to her for clearly shouldering more than her share of work. We are also fortunate that Dr Wiegold who came to visit us in Canberra primarily to stimulate his scholarly work has willingly and ably taken on large teaching responsibil­ities to the detriment of his research. Since it seems likely that the department will acquire at most two new posts in the next triennium it will almost certainly be necessary to restrict student numbers in 1971. Preliminary studies towards this are under way.

The changes foreshadowed in last year's report in the first and third year honours courses seem to be having the desired effect.

A brief introduction to computing was introduced into the first year course in the hope that this would prove useful to the students and might interest some of those who do not find the rest of the course to their taste. The main burden of this course fell on Dr Ward who, assisted by Dr Steiger, prepared a manual, gave some introductory lectures, was available for tutorial help throughout the year, and did much of the routine work (such as punching cards). It is clear that quite a number of students got considerable value from the course and also that quite a few did not. The work spent on the latter seems wasted, next year this part of the course will be optional. This together with some part-time tutoring and the hire of a card punch should reduce the burden on Dr Ward to manageable proportions.

The change to the semester system in the Science Faculty has created some jealousies between Science and Arts students because they are taught together but administered differently. There was also a very marked falling off in the performance of Science students in second term because of semester examinations.

The statistics on student performance which are attached deserve some comment. The enrollment figures for each course represent all students who actually attempted the course; they are therefore somewhat lower, especially for second semester courses, than those available from the administration. The second year results are disappointing. They are partly explained by the fact that quite a number of students attempted the course after having been advised that their first year results did not really warrant this. However even among students whose first year results appeared satisfactory there was a failure-plus-wastage rate of about 30%. An important factor in this seems to be the able but poorly motivated student wiose ability carries him safely through first year but who is unprepared to meet the greater demands for steady work made by the second year course. The first of these factors (ignoring advice) could Jperhaps be met by restricting entry into the second year.

We are indebted to the Department of Mathematics for allowing some of our fourth year students to attend courses and in particular to Dr Edwards for supervising the project of one of them.

Postgraduate At the beginning of the year there were eight postgraduate students

(6 Masters, 2 Ph.D.). During the year four more started work (2 Masters, 2 Ph.D.). Three students successfully completed M.Sc. courses and one did not submit a thesis. There was, as usual, exchange of supervision with the Department of Mathematics; Mr Coppel supervised one of our students; Professor Neumann, Dr Wiegold and Dr Cossey helped supervise their students.

55/1970

3.

Student Participation Informal methods of connnunication have always existed and been

largely successful. There has, I believe and regret, been some falling off in this sort of activity as student numbers have increased.

A widely publicized general meeting of students and staff to discuss formal methods of communication was held in second term. About 30 students (out of about 450) .and 10 staff attended. It was decided to hold general meetings once a term and to form a student-staff committee consisting of all the staff and an equal number of students drawn from every unit offered. The committee has met twice and had useful discussions. A general meeting was held in third term; it was poorly attended (about 10 students). Students participated usefully in the post-mortems that were conducted on this year's courses.

Research The department acted as host to the Ninth Summer Research Institute

of the Australian Mathematical Society. Professor Neumann was Director, Dr Dey, Secretary and Dr Christensen, Treasurer. The galaxy of visitors attracted a large participation to make this the biggest (and best?) institute so far. The department is grateful to the university for the assistance if provided for the institute.

There continues to be active research work with the main emphasis on group theory. Various members of the department were invited during the year to talk about their work: Professor Neumann at Monash and Flinders; Dr Wiegold at Newcastle, Monash and Sydney; Dr Cossey at Monash; Dr Donaldson at Macquarie.

Other Activities Professor Neumann and I continue our activities concerned with school

education through the Australian Association of Mathematics Teachers and its local branch. Professor Neumann gave a series of lectures to the Sunnner School for talented high school pupils held in January.

Notes on Pure Mathematics 2 and 3 appeared and 4 would have but for production delays.

M.F. Newman Acting Head

-Publications 4.

Baker, J.W.* and Neal J. Rothman +

Christensen, c.

Cossey, P.J. **

Cossey, P,J. **

; ; Brady, J,M. , R,A. Bryce and P.J. Cossey**

+ Oates-Macdonald, Sheila, Anne Penfold Street and P.J. Cossey**

Cossey, P.J,** and Alice Whittemore+.

Dey, I,M,S.

Dey, I,M,S.

Howe, A. 0

Neumann, Hanna

Gupta, C,K,*, N,D. Gupta* and M.F. Newman

Steiger, W,L.

key

+

55/1970

Separating points by semicharacters in topological semigroups. Proc.Amer.Math.Soc.~ 235-239(1969).

A basis for the laws of the variety s~30 • J.Austral.Math.Soc,,10(1969),

Critical groups and the lattice of varieties, Proc.Amer.Math,Soc, ~ 217-221(1969).

Some classes of indecomposable varieties of groups. J.Austral.Math.Soc • .2., 387-398(1969).

On certain abelian-by-nilpotent varieties. Bull.Austral.Math,Soc, l, 403-416(1969).

On the laws of certain linear groups, Bull. Amer,Math,Soc, 11, 361-363(1969).

On the Frattini subgroup, Proc.Amer,Math,Soc, 21, 699-702(1969).

Free products and residual nilpotency. Bull,Austral.Math,Soc, l, 11-13(1969).

Embeddings in non-Hopf groups. J.London Math,Soc, (2), l, 745-749(1969).

Holomorphicity of Perturbed Canonical Systems with Periodic Coefficients. J.Differential Equations; ~ 585-591(1969).

Schwartz distributions, Notes on Pure Mathematics 3, 1969, (x) + 95pp. ANU Press.

Some finite nilpotent p-groups, J,Austral.Math,Soc,.2_, 287-288(1969)

Some Kolmogoroff-type inequalities for bounded random variables, Biometrika, 54, 641-647(1967), [Correction to : ibid,,~ 433(1968)]

* Former member. Based on work done while a member of the Department,

+ ** ; 0

Not a member of this University. Based on work done prior to joining this University. A Member of the Department of Mathematics, I.A.s. Based on work done while a member of the Department of Mathematics, I.A.s.

..

5. Steiger, W.L.

Steiger, W.L.

Steiger, w.L.

Ward, M.A.

Havas, G.* and M.A. Ward

Wtegold, J.

Wiegold, J.

:t: :t: Leadham-Green, C.R. , Peter M. HeUtllann and J. Wiegold

key

55/1970

A best possible Kolmogoroff-type inequality for martingales and a characteristic property, Ann.Math. Statist., ~ 764-769(1969).

On a generalization of the Cauchy­Schwartz inequality. Amer.Math. Monthly, 76 815-816(1969).

A probability inequality. J.Math.(Lahore) b 39-41(1969).

Basic commutators. Philos.Trans. Roy.Sac.London Ser.A., 264, 343-412(1969).

Lattices with sublattices of a given order. J.Combinatorial Theory, 1, (1969).

Ext(Q,Z) is the additive group of real numbers. Bull.Austral.Math. Soc.!., 341-343(1969).

Schunk classes are nilpotent product closed. Bull.Austral.Math.Sac. !., 27-28(1969).

The breadth and the class of a finite p-group. J.London Math.Soc. (2), !., 409-420(1969).

* Former member. Based on work done while a member of the Department.

+ Not a member of this University.

THE AUdTRALIAN NATIONAL UNIVERSITY ,

DEPARTr&.."'NT OF PURE MATHEMATICS ANALYSIS OF STUDENT PERFOillMNCE

Percent of Number Enrolled Percent of Number Sitting

1 2 3 4 5 6 7 8 9 10 11 Wastage

Subject or unit Enrolled 17astage plus Sitting IIl) D CREDIT PM PASS FAIL failure

General Mathematics 77 ( 1 ocy;) 11 ( 147';) 26 ( 345'~) 66(86%) n;a n/a 12(18%) n/a 39(58%) 15( 235'S) Pure Mathematics I 196( 1 OO>;) 31(16o/;) 68 ( 3570) 165(04:/4) n/a 2 ( 1;;) 16(10%) n/a 11O(677o) 37(22~;) Pure ifathematics I(H) 44~ 1 OO'i-:~ 1 ~ 250) 3F1/;) 43f98%) 14(33"j) 18(42%) 9(21%) n/a 0 2(5%) Pure Mathematics II 44 1O0/o 16 36%) 24 55o/;) 28 64%) n/a 0 2~7%) n/a 18( 64%) 8(29%) Pure Mathematics IIA 31 ( 100'.%) 7(23"/o) 9(29%) 24( 777; ) n/a 0 4 17~~) n/a 18(75%) 2 ( 8c-,;) Pure Mathematics IIB 19( 100Jo) 0 2( 11%) 19(1O0/o) n/a 0 T%) n/a 16( 84%) 2(11%) Pure Mathematics II(H) 9( 100%) 2(227;) 2(22%) 7(7eyo) 1 ( 14%) 4( 57~;) 2 29%) n/a 0 0 Pure Mathematics IIA(H) 1 3~ 1 OOj'.; ) 0 0 13(1oajo) 4( 31%) 5(38%) 3 23%) n/a 1 (8%) 0 Pure Mathematics IIB(H) 12 100%) 0 0 121100)b) 3( 257';) 4(33%) 5(42%) n/a 0 0 Pure Mathematics IIC(H) 11 ( 1OCt;;) 0 0 11 100%) 5( 45o/; ) 5(45%) 1 ~9%) n/a 0 0 Pure Mathematics III 14( 1 ooi;) 4( 29r;) 4~297; ) 1 O 711;) n/a 0 1 10%) n/a 9(90%) 0 Pure Mathematics IIIA 14~ 1OCrfa) 1 (7%) 1 7%) 13~937;) n/a 1 (8%) 2~15%~ n/a 1op1%~ ) Pure Mathematics IIIB 13 10ay;) 0 0 13 100/o) n/a 0 5 3f;Jo n/a 8 62% 0 Pure Mathematics III(H) 3 1 1 2 0 1 1 n/a 0 0 Pure Mathematics IIIA(H) 7 1 1 6 5 0 1 n/a 0 0 Pure Mathematics IIIB(H) 6 0 0 6 5 1 0 n/a 0 0 Pure Mathematics IIIC(H) 5 0 0 5 4 1 0 n/a 0 0 ==================-======-===-=====-=---------------------------------------------------------------------------------------------- -----------

Enrolled Withdrawn Part-time Sitting I Pure Mathematics IV 5 1 1 3 3 Master's Enrolled (30 June I 69) 5

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Enrolled (30 June 169) 4 \J1 Ph.D. '--... n/a = not available

->.

\.0 -.J HD = High Distinction 0

D = Distinction PM = Pass with Merit